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On a Timoshenko system with thermal coupling on both the bending moment and the shear force

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Abstract

The Timoshenko system is a very well-known model for vibrations of elastic beams, which is given by the coupling of two forces acting on the system: the shear force and the bending moment. In the non-isothermal case, that is, when the model is subject to the temperature variation, we consider the thermal effect acting on the whole system, that is, we propose a new thermoelastic Timoshenko system by coupling thermal laws on both the shear force and the bending moment under the Fourier’s law. Then, we show that such a fully thermoelastic system is exponentially stable without assuming equal wave speeds and also independent of any boundary conditions.

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Acknowledgements

The authors are very grateful to: (i) the anonymous referee for raising questions on “temperatures”, which made the authors improve (and correct) the previous version of the article up to the present one; (ii) the professors Jaime E. Muñoz Rivera, Ramon Quintanilla, and Reinhard Racke, for all helpful discussions and careful considerations on the modeling in thermoelasticity; (iii) the collaboration of Sandro B. Pinheiro (currently at the State University of Maringá, Brazil, as a Ph.D. student) for kindly dedicating time to draw Pictures 1 and 2.

Funding

A. H. Caixeta has been partially supported by the Brazilian Agency CAPES within the project “PNPD - UEL/Matemática Aplicada e Computacional” Scholarship Number 1622327. M. A. Jorge Silva has been partially supported by the CNPq Grant 441414/2014-1 and Fundação Araucária Grant 066/2019. Research of D. S. Almeida Júnior has been partially supported by the CNPq Grants 458866/2014-8 and 310423/2016-3, and PNPD/CAPES/INCTMAT/LNCC Grant 88887.351763/2019-00.

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Authors and Affiliations

  1. Department of Mathematics, State University of Londrina, Londrina, PR, 86057-970, Brazil

    M. O. Alves, A. H. Caixeta, M. A. Jorge Silva & J. H. Rodrigues

  2. Department of Mathematics, Federal University of Pará, Belém, PA, 66075-110, Brazil

    D. S. Almeida Júnior

Authors
  1. M. O. Alves
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  2. A. H. Caixeta
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  3. M. A. Jorge Silva
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  4. J. H. Rodrigues
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  5. D. S. Almeida Júnior
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Correspondence to M. A. Jorge Silva.

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Dedicated to Professor Jaime E. Muñoz Rivera on the occasion of his 60th birthday

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Alves, M.O., Caixeta, A.H., Jorge Silva, M.A. et al. On a Timoshenko system with thermal coupling on both the bending moment and the shear force. J. Evol. Equ. 20, 295–320 (2020). https://doi.org/10.1007/s00028-019-00522-8

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